theoLmoms {lmomco} | R Documentation

## The Theoretical L-moments and L-moment Ratios using Integration of the Quantile Function

### Description

Compute the theoretrical L-moments for a vector. A theoretrical L-moment in integral form is

λ_r = frac{1}{r} sum^{r-1}_{k=0}{(-1)^k {r-1 choose k} frac{r!:I_r}{(r-k-1)!k!} } mbox{, in which }

I_r = int^1_0 X(F) times F^{r-k-1}(1-F)^{k},mathrm{d}F mbox{,}

where X(F) is the quantile function of the random variable X for nonexceedance probability F, and r represents the order of the L-moments. This function actually dispatches to `theoTLmoms` with `trim=0` argument.

### Usage

    theoLmoms(para,nmom=5,verbose=FALSE)

### Arguments

`para` | A distribution parameter object of this package `vec2par`.
`nmom` | The number of moments to compute. Default is 5.
`verbose` | Toggle verbose output. Because the R function `integrate` is used to perform the numerical integration, it might be useful to see selected messages regarding the numerical integration.

### Value

An R `list` is returned.

`lambdas` | Vector of the TL-moments. First element is λ_1, second element is λ_2, and so on.
`ratios` | Vector of the L-moment ratios. Second element is tau_2, third element is tau_3 and so on.
`trim` | Level of symmetrical trimming used in the computation, which will equal zero (the ordinary L-moments).
`source` | An attribute identifying the computational source of the L-moments: “theoTLmoms”.

### Note

The actual function used is `theoLmoms(para,nmom=nmom,trim=0,verbose=verbose`.

### Author(s)

W.H. Asquith

### References

Hosking, J.R.M., 1990, L-moments—Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, vol. 52, p. 105–124.

### See Also

`theoTLmoms`, `par2qua`, `TLmoms`, `lmom.ub`

### Examples

    para <- vec2par(c(0,1),type='nor') # standard normal
    TL00 <- theoLmoms(para) # compute ordinary L-moments

[Package lmomco version 0.92 Index]